Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, eg. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple $(S,L,H)$. Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.Comment: 60 pages, 14 figures. We are still interested in receiving correction

We propose an approach to quantum error correction based on coding and continuous syndrome readout via scattering of coherent probe fields, in which the usual steps of measurement and discrete restoration are replaced by direct physical processing of the probe beams and coherent feedback to the register qubits. Our approach is well matched to physical implementations that feature solid-state qubits embedded in planar electromagnetic circuits, providing an autonomous and "on-chip" quantum memory design requiring no external clocking or control logic.

Quantum computation requires qubits that satisfy often-conflicting criteria, including scalable control and long-lasting coherence [1]. One approach to creating a suitable qubit is to operate in an encoded subspace of several physical qubits. Though such encoded qubits may be particularly susceptible to leakage out of their computational subspace, they can be insensitive to certain noise processes [2, 3] and can also allow logical control with a single type of entangling interaction [4] while maintaining favorable features of the underlying physical system. Here we demonstrate a qubit encoded in a subsystem of three coupled electron spins confined in gated, isotopically enhanced silicon quantum dots [4, 5]. Using a modified "blind" randomized benchmarking protocol that determines both computational and leakage errors [6, 7], we show that unitary operations have an average total error of 0.35%, with 0.17% of that coming from leakage driven by interactions with substrate nuclear spins. This demonstration utilizes only the voltage-controlled exchange interaction for qubit manipulation and highlights the operational benefits of encoded subsystems, heralding the realization of high-quality encoded multi-qubit operations [4, 8].Electrons trapped in silicon heterostructures have many attractive features, including very long coherence times in isotopically enriched material [9, 10] and compatibility with standard fabrication techniques. Singlespin qubits have recently demonstrated high-fidelity RFcontrolled single-qubit operations [10, 11] and two-qubit gates using the exchange interaction [12][13][14]. However, using RF signals for single-qubit control requires a large, stable magnetic field and introduces challenges with crosstalk. Fortunately, electron spins are particularly well-suited to forming encoded qubits. Two coupled electron spins can be operated at near-zero magnetic field as a "singlet-triplet" qubit [15,16]. That qubit is insensitive to uniform magnetic field fluctuations but still requires a magnetic field gradient for universal control. Three coupled electrons [17] can form a qubit with a tunable electric dipole moment, which could enhance RF selectivity, or the exchange-only qubit, which can be universally controlled using only the exchange interaction and does not require synchronization of gate operations with a local oscillator. Exchange is highly local and can be accurately controlled with a large on-off ratio using only fast voltage pulses. The combination of these features makes the exchange-only qubit especially attractive b X2

We analyze the design of a potential replacement technology for the commercial ferrite circulators that are ubiquitous in contemporary quantum superconducting microwave experiments. The lossless, lumped element design is capable of being integrated on chip with other superconducting microwave devices, thus circumventing the many performance-limiting aspects of ferrite circulators. The design is based on the dynamic modulation of DC superconducting microwave quantum interference devices (SQUIDs) that function as nearly linear, tunable inductors. The connection to familiar ferrite-based circulators is a simple frame boost in the internal dynamics' equation of motion. In addition to the general, schematic analysis, we also give an overview of many considerations necessary to achieve a practical design with a tunable center frequency in the 4-8 GHz frequency band, a bandwidth of 240 MHz, reflections at the -20 dB level, and a maximum signal power of approximately order 100 microwave photons per inverse bandwidth.

We report on the design and performance of an on-chip microwave circulator with a widely (GHz) tunable operation frequency. Nonreciprocity is created with a combination of frequency conversion and delay, and requires neither permanent magnets nor microwave bias tones, allowing on-chip integration with other superconducting circuits without the need for high-bandwidth control lines. Isolation in the device exceeds 20 dB over a bandwidth of tens of MHz, and its insertion loss is small, reaching as low as 0.9 dB at select operation frequencies. Furthermore, the device is linear with respect to input power for signal powers up to hundreds of fW (≈10 3 circulating photons), and the direction of circulation can be dynamically reconfigured. We demonstrate its operation at a selection of frequencies between 4 and 6 GHz.